Conceptual Aspects of Gauge/Gravity Duality

Sebastian de Haro University of Cambridge and University of Amsterdam

Cluster of Excellence »Image, Knowledge, Gestalung. An Interdisciplinary Laboratory« and High-Energy and Mathematical Physics Department

Humboldt Universität, Berlin, 24 June 2015

Based on:

• Dieks, D. van Dongen, J., de Haro, S. (2015). Studies in History and Philosophy of Modern Physics, forthcoming

• de Haro, S. (2015). Studies in History and Philosophy of Modern Physics, forthcoming

• de Haro, S., Teh, N., Butterfield, J. (2015). Philosophy of Science, submitted

Introduction

• In recent years, gauge/gravity dualities have been an important focus in quantum gravity research

•Gauge/gravity dualities relate a theory of gravity in 𝐷 dimensions to a quantum field theory (no gravity!) in 𝐷 − 1 dimensions• Also called ‘holographic’• Not just nice theoretical models: one of its versions

(AdS/CFT) successfully applied in experiments at RHIC (Brookhaven, NY)

2

Motivation: Foundations of Physics

•Gauge/gravity dualities are claimed to describe theories of ‘quantum gravity’• Should provide conceptual insights into the difficult

programme of quantising gravity

•Dualities are themselves of interest, instantiating philosophical debates about theories that ‘say the same thing, in different words’

• It is often claimed that, in these models, space-time and/or gravity ‘disappear/dissolve’ at high energies; and ‘emerge’ in some suitable classical limit• Analysing these claims and clarifying the meaning of

‘emergence’ of space-time/gravity can provide insights into the conditions under which emergence can occur

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Aim and plan of this talk

•To expound on conceptual aspects of the best studied gauge/gravity duality (AdS/CFT), with an eye on foundational and philosophical issues that it bears on:• Provide a definition of ‘duality’ and make two general

comments:i. Duality between two theories as being ‘gauge related’ in the

general philosophical sense of being physically equivalent: ‘saying the same thing, in different words’

ii. On the relation between dualities and gauge symmetries: discussing some subtleties

• Discuss background-independence• Work out an interpretation of this duality• Discuss the relation between duality and emergence of space-

time and gravity4

AdS/CFT: crude description

•AdS/CFT is a relationship (duality) between:• A theory of gravity in anti-de Sitter space (a solution of

Einstein’s equations with a negative cosmological constant): the ‘bulk’; and, in the bulk, a string theory, and its low energy limit, Einstein’s theory of gravity

• A special kind of quantum field theory: a conformal field theory, i.e. a quantum field theory that is invariant under conformal transformations (coordinate transformations that give rise to a scale factor): the ‘boundary’

•Discovered by Maldacena in 1997•Concrete realisation of the general phenomenon of

‘holography’ conjectured by ’t Hooft in 19935

Duality: a simple (even simplistic!) definition

• Regard a theory as a triple ℋ, 𝒬, 𝐷 : states, physical quantities, dynamics• ℋ = states (in Hilbert space)• 𝒬 = operators (self-adjoint, renormalizable, invariant under symmetries)• 𝐷 = dynamics (e.g. Lagrangian and integration measure)

• A duality is an isomorphism (i.e. a structure-preserving bijective map) between two theories ℋ𝐴, 𝒬𝐴, 𝐷𝐴 and ℋ𝐵 , 𝒬𝐵, 𝐷𝐵

• There exist bijections: • 𝑑ℋ: ℋ𝐴 → ℋ𝐵 ,

• 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵

and pairings (expectation values) 𝒪, 𝑠 𝐴 such that:𝒪, 𝑠 𝐴 = 𝑑𝒬 𝒪 , 𝑑ℋ 𝑠

𝐵∀𝒪 ∈ 𝒬𝐴, 𝑠 ∈ ℋ𝐴

• I call this definition ‘simple, even simplistic’ because in gauge/gravity duality the Hilbert space is only known in special limits (e.g. semi-classical limit): it will be somewhat ‘formal’, yet conceptually helpful

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Duality (more refined version)

• Add external parameters 𝒞 (e.g. couplings, sources)

• The theory is given as a quadruple ℋ, 𝒬, 𝒞, 𝐷

• Duality is an isomorphism ℋ𝐴, 𝒬𝐴, 𝒞𝐴 ≃ ℋ𝐵, 𝒬𝐵, 𝒞𝐵 . There exist bijections: • 𝑑ℋ: ℋ𝐴 → ℋ𝐵

• 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵

• 𝑑𝒞: 𝒞𝐴 → 𝒞𝐵

such that:

𝑂, 𝑠 𝑐 ,𝐷𝐴= 𝑑𝒪 𝑂 , 𝑑𝒮 𝑠 {𝑑𝒞(𝑐)} ,𝐷𝐵

∀𝒪 ∈ 𝒬𝐴, 𝑠 ∈ ℋ𝐴, 𝑐 ∈ 𝒞𝐴

𝒪, 𝑠 𝑐 ,𝐷𝐴= 𝑑𝒬 𝒪 , 𝑑ℋ 𝑠

{𝑑𝒞(𝑐)} ,𝐷𝐵(1)

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Duality as a unitary transformation

• One can show that, given the isomorphisms 𝑑ℋ: ℋ𝐴 → ℋ𝐵 and 𝑑𝒬: 𝒬𝐴 → 𝒬𝐵, 𝑑ℋ is in fact a unitary operator:

𝑑ℋ𝑑ℋ† = 𝑑ℋ

† 𝑑ℋ = 𝟙 ; and

𝑑𝒬 𝒪𝐴 = 𝑑ℋ 𝒪𝐴 𝑑ℋ†

• Thus, this is an ordinary unitary transformation on the Hilbert space, at the level of states and observables. This raises the following

• Question: to what degree are the two theories structurally ‘different’?• A priori, they differ in their ‘dynamics’ 𝐷

• But the ‘dynamics’ already seems included in ℋ, 𝒬, 𝒞 : it may be possible to reconstruct a Lagrangian, say, from this data

• So the two theories are really different representations of the same Hilbert space and set of observables: they are ‘theoretically equivalent’

• This is my first comment (i) on ‘duality’: under some plausible assumptions, the two theories are physically equivalent: they ‘say the same thing, in different words’ (different representations) 8

Geometry of AdS𝐷

• Hyperboloid in 𝐷 + 1 dimensions:

−𝑋02 − 𝑋𝐷

2 +

𝑖=1

𝐷−1

𝑋𝑖2 = −ℓ2

• Constraint can be solved introducing 𝐷 coordinates:𝑋0 = ℓ cosh 𝜌 cos 𝜏𝑋𝐷 = ℓ cosh 𝜌 sin 𝜏𝑋𝑖 = ℓ sinh 𝜌 Ω𝑖 𝑖 = 1, … , 𝑑 = 𝐷 − 1 , Ω𝑖 = unit vector

• Leading to: d𝑠2 = ℓ2 − cosh2 𝜌 d𝜏2 + d𝜌2 + sinh2 𝜌 dΩ𝐷−22

• Symmetry group SO 2, 𝑑 apparent from the construction

• Riemann tensor given in terms of the metric (negative curvature):

𝑅𝜇𝜈𝜆𝜎 = −1

ℓ2 𝑔𝜇𝜆𝑔𝜈𝜎 − 𝑔𝜇𝜎𝑔𝜈𝜆

𝑋𝑖

𝑋0

𝑋𝐷

ℓ

9

Geometry of AdS𝐷

• Useful choice of local, Poincaré coordinates:

d𝑠2 =ℓ2

𝑟2 d𝑟2 + 𝜂𝑖𝑗 d𝑥𝑖d𝑥𝑗 , 𝑖 = 1,… , 𝑑 = 𝐷 − 1

• 𝜂𝑖𝑗 = flat metric (Lorentzian or Euclidean signature)

• Fefferman and Graham (1985): for a space that satisfies Einstein's equations with a negative cosmological constant, and given a conformal metric at infinity, the line element can be written as:

d𝑠2 =ℓ2

𝑟2d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖d𝑥𝑗

𝑔𝑖𝑗 𝑟, 𝑥 = 𝑔 0 𝑖𝑗 𝑥 + 𝑟 𝑔 1 𝑖𝑗 𝑥 + 𝑟2𝑔 2 𝑖𝑗 𝑥 + ⋯

• Einstein’s equations now reduce to algebraic relations between 𝑔 𝑛 𝑥 𝑛 ≠ 0, 𝑑 and 𝑔 0 𝑥 , 𝑔 𝑑 𝑥 (de Haro et al. 2001)

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• Corollary: the diffeomorphisms that leave the asymptotic metric fixed reduce to conformal transformations of the induced boundary metric 𝑔 0 𝑖𝑗 𝑥

• This bulk metric is very general: it includes pure AdS, AdS black holes (i.e. any solution with zero stress-energy tensor and negative cosmological constant). AdS/CFT is not restricted to the most symmetric case! (common misconception): hence the name ‘gauge/gravity’

• The above construction generalizes to the case of non-zero stress-energy tensor, i.e. gravity coupled to matter

• Matter field 𝜙 𝑟, 𝑥 (for simplicity, take 𝑚 = 0), solve KG equation coupled to gravity:

𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + 𝑟 𝜙 1 𝑥 + ⋯ + 𝑟𝑑𝜙 𝑑 𝑥 + ⋯

• Again, 𝜙 0 𝑥 and 𝜙 𝑑 𝑥 are the boundary conditions and all other coefficients 𝜙 𝑛 𝑥 are given in terms of them (as well as the metric coefficients)

Adding Matter

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AdS/CFT

•𝐷-dim AdS

•𝐷-dim (AL)AdS

• d𝑠2 =ℓ2

𝑟2 d𝑟2 + 𝑔𝑖𝑗 𝑟, 𝑥 d𝑥𝑖d𝑥𝑗

• 𝑔 𝑟, 𝑥 = 𝑔 0 𝑥 + ⋯ + 𝑟𝑑𝑔 𝑑 𝑥

• Field 𝜙 𝑟, 𝑥 , mass 𝑚

• 𝜙 𝑟, 𝑥 = 𝜙 0 𝑥 + ⋯ + 𝑟𝑑𝜙 𝑑 𝑥

• CFT on ℝ𝐷−1

•QFT with a fixed point• Metric 𝑔 0 (𝑥)

• 𝑇𝑖𝑗 𝑥 =ℓ𝑑−1

16𝜋𝐺𝑁𝑔 𝑑 𝑥 + ⋯

•Operator 𝒪 𝑥 , scaling dimension Δ 𝑚• Coupling 𝜙 0 𝑥

• 𝒪 𝑥 = 𝜙 𝑑 𝑥

Fields Operators

Normalizable mode (sub-leading) Exp. val. (state)

Non-normalizable mode (leading) Coupling 12

Example: AdS5 × 𝑆5 ≃ SU 𝑁 SYM

AdS5 × 𝑆5

• Type IIB string theory

• Limit of small curvature:supergravity (Einstein’s theory + specific matter fields)

• Symmetry of AdS: diffeo’s that preserve form of the metric generate conformal transformations on the bdy

• Symmetry of 𝑆5

SU 𝑁 SYM

• Supersymmetric, 4d Yang-Mills theory with gauge group SU(𝑁)

• Limit of weak coupling: ’t Hooft limit (planar diagrams)

• Classical conformal invariance of the theory

• Symmetry of the 6 scalar fields

• Limits are incompatible (weak/strong coupling duality: useful!)• Only gauge invariant quantities (operators) can be compared• Symmetry:

SO 2,4 × SO 6

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AdS/CFT Duality

• AdS/CFT can be described in terms of the quadruple ℋ, 𝒬, 𝒞, 𝐷 : • Normalizable modes correspond to exp. vals. of operators (choice of state)

• Fields correspond to operators

• Boundary conditions (non-normalizable modes) correspond to couplings

• Formulation otherwise different (off-shell Lagrangian, different dimensions!)

• Two salient points of :• Physical quantities, such as boundary conditions, that are not determined by

the dynamics, now also agree: they correspond to couplings in the Lagrangian. This is the case in any duality that involves parameters that are not expectation values of operators, e.g. T-duality (𝑅 ↔ 1/𝑅), electric-magnetic duality (𝑒 ↔ 1/𝑒)

• It is also more general: while ℋ, 𝒬, 𝐷 are a priori fixed, 𝒞 can be varied at will (Katherine Brading: ‘modal equivalence’). We have a multidimensional space of theories

• Dualities of this type are not isomorphisms between two given theories, but between two sets of theories

ℋ

𝒬

𝒞

𝐷

(1)

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AdS/CFT Duality (Continued)

•String theory in (AL)AdS space = QFT on boundary•Formula 1 is generated by:

𝑍string 𝜙 0 = 𝜙 0,𝑥 =𝜙 0 𝑥

𝒟𝜙 𝑒−𝑆 𝜙 = exp d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

•The correlation functions of all operators match•Physical equivalence, mathematical structure different•Large distance ↔ high energy divergences•Strictly speaking, the AdS/CFT correspondence has the

status of a ‘conjecture’, though there is strong evidence for it: and it is usually called a ‘correspondence’: compare e.g. with Fermat’s last ‘theorem’ before it was proven!

(1′)

𝑐𝐵 = 𝑑𝒞 𝑐𝐴

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Conditions for AdS/CFT Duality

• What could lead to the failure of AdS/CFT as a duality?

• Two conditions must be met for this bijection to exist. The observable structures of these theories should be:

i. Complete (sub-) structures of observables, i.e. no other observables can be written down than (1): this structure of observables contains what the theories regard to be ‘physical’ independently on each side of the duality

ii. Identical, i.e. the (sub-) structures of observables are identical to each other

If ii. is not met, we can have a weaker form of the conjecture: a relation that is non-exact. For instance, if the duality holds only in some particular regime of the coupling constants

• There are no good reasons to believe that i. fails

• Whether ii. is met is still open, but all available evidence indicates that it is satisfied, including some non-perturbative tests (protected quantities: e.g. Green (1999))

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Invisibility

• Now for my second comment (ii) on the relation between ‘duality’ and ‘gauge-invariance’: the two theories are ‘closer than one might think’

• Gauge/gravity duality relates ‘gauge-invariant’ quantities only

• Horowitz and Polchinski (2006) introduce the notion of ‘invisibility’: in AdS/CFT, the diffeomorphism symmetry of the bulk theory is ‘invisible’ to the boundary theory, thus showing that diffeomorphism symmetry (and relatedly, space-time) is an ‘emergent’ property• Invisible diffeomorphisms: bulk diffeomorphisms that restrict to the identity

map at the boundary, hence do not change boundary quantities• Visible diffeomorphisms: those that do not restrict to the identity map on the

boundary

• But one can show that, once a solution is fixed, there are no invisible diffeomorphisms in (AL)AdS!

• The only non-trivial diffeomorphisms are of PBH (“Penrose-Brown-Henneaux”) type: but these are visible! They reduce to conformal transformations on the boundary (corollary to Fefferman-Graham)

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Sketch of the proof of the corollary

• Fefferman and Graham (1985) showed that, for a space that satisfies Einstein's equations with a negative cosmological constant, and given a conformal metric at infinity, the line element can be written in the following form:

d𝑠2 =ℓ2

4𝜌2d𝜌2 +

ℓ

𝜌𝑔𝑖𝑗 𝜌, 𝑥 d𝑥𝑖d𝑥𝑗

where 𝑔𝑖𝑗 𝜌, 𝑥 is a regular function, inducing the conformal boundary metric 𝑔 0 𝑖𝑗 𝑥 ≔ 𝑔𝑖𝑗 0, 𝑥

• Consider diffeomorphisms: 𝜌 = 𝜌 𝑒−2𝜉 𝑥 , 𝑥𝑖 = 𝑥𝑖 + 𝜉𝑖 𝜌, 𝑥

• Condition that they leave the boundary form of the solution fixed:

𝛻𝑖𝜉𝑗 𝑥 + 𝛻𝑗𝜉𝑖 𝑥 −2

𝑑𝑔 0 𝑖𝑗 𝑥 𝛻𝑘𝜉𝑘 𝑥 = 0 𝜉𝑖 𝑥 ≔ 𝜉𝑖 0, 𝑥

• This is the condition for an infinitesimal boundary coordinate transformation 𝜉𝑖 𝑥 to give a scale transformation: they generate the boundary conformal group (de Haro (2016))• This argument extends to the non-linear level

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Visibility

• However, if one requires that the boundary coordinate transformation vanish, 𝜉𝑖 𝑥 = 0, then the diffeomorphism is in fact identically zero throughout the bulk. Thus there are no invisible diffeomorphisms: the two theories are ‘closer than one might think’

• There are only visible diffeomorphisms, and these are the PBH ones (suitably generalised to higher dimensions)• They are visible because they induce conformal transformations on the

boundary

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Remarks on Background-Independence

• Theories of gravity are usually required to be ‘background-independent’. In Einstein’s theory of relativity, the metric is a dynamical quantity, determined from the equations of motion rather than being fixed from the outset

• A common criticism of string theory is that it is ‘background dependent’: Smolin (2005)

• The concept of ‘background-independence’ does not have a fixed meaning, but comes in degrees: see Belot (2011)

• Here I will adopt a ‘minimalist approach’: a theory is background independent if it is generally covariant and its formulation does not require the introduction of a background/fixed metric field. In particular: the metric has relevant degrees of freedom that are determined dynamically from the equations of motion

• In this minimalist sense, classical gravity in AdS is fully background independent: Einstein’s equations with negative cosmological constant• Quantum corrections do not change this conclusion: they appear

perturbatively as covariant higher-order corrections to Einstein’s theory

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Remarks on Background-Independence

• Could background-independence be broken by the asymptotic form of the metric?• This criticism has been aired by Smolin (2005), and (more tentatively) by Rickles

(2013). It is implicitly endorsed in Hubeny’s (2015) recent review of AdS/CFT

• The asymptotic form of the metric is determined by a choice of boundary condition. The equations of motion do not determine them: they need to be specified additionally (de Haro et al. 2001)

• But this is not a restriction on the class of solutions considered; as in classical mechanics, the laws (specifically: the equations of motion) simply do not contain the information about the boundary/initial conditions• We do not say that classical mechanics is dependent on a fixed coordinate, or is

parametrisation-dependent, just because initial conditions are required to pick particular solutions of the equations of motion. Any theory described by differential equations will require such choices of initial or boundary conditions

• Boundary conditions do not need to preserve the symmetries of the laws. Thus this does not seem to be a case of lack of background independence of the theory

• At most, it may lead to spontaneous breaking of the symmetry in the sense of: a choice of a particular solution

• Hence, the background-independence of the theory is well established21

Diffeomorphism Invariance of (1’)

• I have discussed background independence at the level of the equations of motion. What about the observables?

• Partition function (1’):• It depends on the boundary conditions on the metric (as do the classical

solutions)• It is diffeomorphism invariant, for those diffeomorphisms that preserve the

asymptotic form of the metric

• Other observables obtained by taking derivatives of (1): they transform as tensors under these diffeomorphisms. These observables are covariant, for odd d (=boundary dimension):

• For odd 𝑑: • Invariance/covariance holds

• For even 𝑑:• Bulk diffeomorphisms that yield conformal transformations of the boundary

metric are broken due to IR divergences (holographic Weyl anomaly). Is this bad?

𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

(1′)

22

Diffeomorphism Invariance (even 𝑑)

• The breaking of diffeomorphism invariance exactly mirrors the breaking of conformal invariance by quantum effects in the CFT

• The partition function now depends on the representative of the conformal structure picked for regularization

• The observables (1’) such as the stress-tensor no longer transform covariantly, but pick up an anomalous term

• Anomalies are usually quantum effects, proportional to ℏ. Here, the anomaly is (inversely) proportional to Newton’s constant 𝐺

• The anomaly is robust: fully non-linear (does not rely on linearization about a particular solution), does not rely on classical approximation to the bulk geometry

• This anomaly does not lead to any inconsistencies because the metric is not dynamical in the CFT: the classical symmetry of the path integral need not be preserved at the quantum level• This may be problematic for some of the applications one normally has in mind in a

theory of quantum gravity

• It is a different issue from the dependence on the asymptotic form of the metric!23

Philosophical Questions

• Is one side of the duality more fundamental?• If QFT more fundamental, space-time could be ‘emergent’

• If the duality is only approximate: room for emergence (e.g. thermodynamics vs. atomic theory):• This is the case in practical applications of AdS/CFT so far

(condensed matter systems, RHIC)

• If duality holds good: one-to-one relation between the values of physical quantities. In this case there are two possible physical interpretations

24

http://math.ucr.edu/home/baez/escher.gifhttp://mathworld.wolfram.com/images/eps-gif/HyperbolicTessellation_1000.gif

• Two-dimensional beings whose rods expand/contract with temperature

• We can deceive them into thinking they live on the Lobachevski plane: non-Euclidean geometry

𝑇 = 0

𝑇 = 𝑅2

25

Poincaré’s parable

Poincaré’s parable

• Poincaré: there is no way to distinguish between the two situations. What is to count as the difference between a real non-Euclidean geometry and a Euclidean world with distorting fields?

• Nothing in the fact of the matter determines which hypothesis is ‘correct’

• He argues that the two theories are equivalent: theoretical equivalence

26

Interpretation

•External point of view: the meaning of the observables is externally fixed. Duality relates different physical quantities• There is no empirical equivalence; since the numbers

correspond to different physical quantities• The symmetry of the terms related by duality is broken by

the different physical interpretation given to the symbols• Example: 𝑟 fixed by the interpretation to mean ‘radial

distance’ in the bulk theory. In the boundary theory, the corresponding symbol is fixed to mean ‘energy scale’. The two symbols clearly describe different physical quantities. More generally, the two theories describe different physics hence are not empirically equivalent

• Only one of the two sides provides a correct interpretation of empirical reality

27

Interpretation• Internal point of view:

• The meaning of the symbols (observables) is not a priori fixed. It is only fixed by matching numbers: between observables and observational outcomes (e.g.: fitting of parameters in inflationary models!)

• There is one single set of observables (physical quantities) that the two theories describe: by definition of duality as isomorphism

• Because the meaning of the symbols is not a priori fixed, the two theories give equivalent descriptions of the same physics (same physical quantities)

• Even though (based on past conventions) the symbols in the two theories seem to have different interpretations (𝑟 as ‘radial distance’ or as ‘energy scale’), they really describe the same physics because they both describe the same observations equally well. There is no distinction between the two

• No possible observation could tell one theory from the other: each observation fits with either of the two theories

• On this view, we would say that we have two formulations of one theory, not two different theories

28

Interpretation

•One formulation may be superior on pragmatic grounds (e.g. computational simplicity in a particular regime)

•The internal point of view seems more natural for theories of the whole world• There are no quantities external to the theory (coupling to

external systems, probes) that could fix an interpretative context• This is supported by the remark that duality is a unitary

transformation: there is no compelling reason why one representation should be superior than another

•Even if one views a theory as a partial description of empirical reality, in so far as one takes it seriously in a particular domain of applicability, the internal view seems the more natural description• Compare: equivalence of frames in special relativity

• Position/momentum duality in quantum mechanics 29

Back to diffeomorphism invariance

• The boundary diffeomorphisms 𝜉𝑖 𝑥 (conformal transformations) represent overall transformations of the correlation functions

• On the internal point of view, two states of the universe related by such transformations are physically equivalent: these transformations count as gauge symmetries

• The external point of view interprets (1’) as coupled to an already interpreted physical background 𝜙 0 𝑥 : in that case, diffeomorphisms become physical

30

𝑍string 𝑟Δ −𝑑𝜙 𝑟, 𝑥𝑟=0

= 𝜙 0 𝑥 = 𝑒 d𝑑𝑥 𝜙 0 𝑥 𝒪 𝑥

CFT

(1′)

Emergence

• An argument along the following lines is often made:

a) Theory F (‘fundamental’) and theory G (‘gravity’) are dual to one another

b) Theory F does not contain gravity (and/or space-time) whereas theory G does

c) Therefore space-time (and/or gravity) emerges in theory G. Theory F is to be regarded as more fundamental

31

•But this argument is problematic: it replaces ‘duality’ by ‘emergence’•Duality is symmetric : if F dual to G, then G is dual to F• Emergence is asymmetric: if F emerges from G, then G

cannot emerge from F; and non-reflexive: F cannot emerge from itself!•We need to explain what produces the asymmetry. In

other words, if F and G are dual to one another, an additional criterion is required for emergence• Emergence of space-time requires more than simply ‘the

space-time being dual to something that is not spatio-temporal’• It might lead to bad heuristics for constructing new

theories, e.g. when the argument is taken as a reason not to pursue theory G but to just work on theory F

• In AdS/CFT: only possible emergence of gravity together with one, spatial dimension• This is a non-trivial task: for obtaining the right classical

dynamics for the metric is hard!32

• On the external view, the two theories describe different physics• The dual theory is only a tool that might be useful, but does not describe

the physics of our world• Here, the idea of ‘emergence’ does not suggest itself: because whichever

side describes our world, it does not emerge from something else• In RHIC-type experiments, there might be ‘emergence’ purely as meaning:

an ‘effective’ space-time. But the space-time has no physical reality on this view (whether the duality is exact or not)

• On the internal view, there is a one-to-one relation between the values of physical quantities• Emergence again does not suggest itself: the two descriptions are

equivalent (the relation is symmetric)

• There may be emergence of (some aspects of) space-time if there is coarse-graining. This, though, is not emergence of one theory from its dual, but emergence restricted to one of the two sides

• If the duality is only approximate then there is clear scope for emergence of space-time (analogy: thermodynamics vs. statistical mechanics)

Emergence

33

Analysing Emergence

• Imagine we have holography and we have coarse-graining: a thermodynamic limit, such as large 𝑁, 𝑉

• In the AdS bulk, such a limit is provided by the limit of small curvature (small cosmological constant)

• In the CFT, this corresponds to the limit of large 𝑁 and weak coupling (so-called ’t Hooft coupling) 34

• The holographic relation may well be a bijective map• There is no reason in this case to think that one side is

more fundamental than the other (left-right)• But the thermodynamic limit introduces the emergence

of gravity in an uncontroversial sense (top-bottom). Call this ‘bulk-bulk’ or ‘boundary-boundary’ emergence

Does Gravity Emerge for Exact Dualities?

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Emergence with Approximate Duality

• If holography only holds approximately, then we do haveemergence of gravity from the microscopic boundary theory

• This is the only way that the strong kind of emergence, sometimes claimed in the literature, can be construed within such a framework. Call this boundary-bulk emergence 36

Emergence of Space and Gravity

• Gravity could thus emerge, in the boundary-bulk sense, in other situations where gauge/gravity duality does not hold exactly (e.g. cosmological scenarios: dS/CFT)

• Bulk-bulk emergence can be applied to AdS/CFT with coarse graining over high-energy degrees of freedom (renormalization group flow)

• In this case, Einstein gravity may emerge from the fundamental bulk theory, whether the latter contains gravity or not

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Conclusions: Gauge/Gravity Duality

• Dual theories are ‘closer than one would expect’i. A duality is a unitary transformation: the two theories ‘say the same thing, in

different words’ii. The notion of invisible diffeomorphisms does not hold up to scrutiny in AdS/CFT• The diffeomorphism structure is visible and mapped, by duality, to the boundary

conformal group

• AdS/CFT satisfies the requirement of background-independence (minimally construed), except:• There is an anomaly in odd boundary dimensions, mirrored by the CFT’s conformal

anomaly

• Interpreting gauge/gravity duality, the internal point of view is the more natural one• In that case, duality amounts to full physical equivalence

• On the internal view, the boundary conformal transformations are ‘gauge’; and so at least some of the gauge structure of the duality is visible

• AdS/CFT is a useful model to address foundational questions that can help study more realistic models such as dS/CFT• In that case, the ‘emergent direction’ is time not space!

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Conclusions: Dualities and Emergence

• In gauge/gravity scenarios with an exact duality, the boundary theory is not necessarily more fundamental than the bulk theory•Duality interpreted in terms of external vs. internal

perspectives, a version of Poincaré conventionalism naturally applies to cosmological models• The bulk does not emerge from the boundary in

such cases

•There are possibilities for emergence (connected with robustness and novelty of behaviour) of one space dimension, and of gravity, in a coarse-graining limit (bulk-bulk, boundary-bulk)

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Thank you!

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